These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.
Have a go at this 3D extension to the Pebbles problem.
This jar used to hold perfumed oil. It contained enough oil to fill granid silver bottles. Each bottle held enough to fill ozvik golden goblets and each goblet held enough to fill vaswik crystal. . . .
A box has faces with areas 3, 12 and 25 square centimetres. What is the volume of the box?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
An irregular tetrahedron has two opposite sides the same length a and the line joining their midpoints is perpendicular to these two edges and is of length b. What is the volume of the tetrahedron?
In the ancient city of Atlantis a solid rectangular object called a Zin was built in honour of the goddess Tina. Your task is to determine on which day of the week the obelisk was completed.
An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?
What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?
Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.
What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?
Here's a chance to work with large numbers...
Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.
How can you change the surface area of a cuboid but keep its volume the same? How can you change the volume but keep the surface area the same?
A plastic funnel is used to pour liquids through narrow apertures. What shape funnel would use the least amount of plastic to manufacture for any specific volume ?
Can you work out the dimensions of the three cubes?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
The builders have dug a hole in the ground to be filled with concrete for the foundations of our garage. How many cubic metres of ready-mix concrete should the builders order to fill this hole to. . . .
What happens to the area and volume of 2D and 3D shapes when you enlarge them?
Practice your skills of measurement and estimation using this interactive measurement tool based around fascinating images from biology.
A right circular cone is filled with liquid to a depth of half its vertical height. The cone is inverted. How high up the vertical height of the cone will the liquid rise?
According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have. . . .
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
Analyse these beautiful biological images and attempt to rank them in size order.
If I don't have the size of cake tin specified in my recipe, will the size I do have be OK?
Can you sketch graphs to show how the height of water changes in different containers as they are filled?
Can you draw the height-time chart as this complicated vessel fills with water?
If the radius of the tubing used to make this stand is r cm, what is the volume of tubing used?
My measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.
Various solids are lowered into a beaker of water. How does the water level rise in each case?
Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?